MAT3333
Course name
MAT 3333 Fundamentals of Analysis and Topology.
Catalog entry: MAT 3333 Fundamentals of Analysis and Topology. Prerequisite: MAT 3003 Discrete Mathematics, or consent of instructor. Topological notions in the real line and in metric spaces. Convergent sequences. Continuous functions. Connected and compact sets. The Intermediate Value and Extreme Value theorems. Sequential compactness and the HeineBorel Theorem.
Prerequisites: MAT 1224 and MAT 3003.
Sample textbooks:
 John M. Erdman, A Problems Based Course in Advanced Calculus. Pure and Applied Undergraduate Texts 32, American Mathematical Society (2018). ISBN: 9781470442460.
 JyhHaur Teh, Advanced Calculus I. ISBN13: 9798704582137.
Topics List
(Section numbers refer to Erdman's book.)
Week  Sections  Topics  Student Learning Outcomes 

13 
Chapters 1 & 2. Appendices C, G & H. 
Operations, order and intervals of the real line. Completeness of the real line. Suprema and infima. Basic topological notions in the real line. 

4 
Chapter 3 
Continuous functions on ℝ. 

45 
Chapter 4 
Convergence of real sequences. The Cauchy criterion. Subsequences. 

6 
Chapter 5 
Connectedness and the Intermediate Value Theorem 

7 
Chapter 6 
Compactness and the Extreme Value Theorem. 

8 
Chapter 7 
Limits of real functions. 

9 & 10 
Chapters 9, 10, 11 
The topology of metric spaces. 

11 
Chapter 12 
Sequences in metric spaces. 

12 
Chapter 14 
Continuity and limits. 

13 
15.1–15.2 
Compact metric spaces. 

14 
Chapter 16 
Sequential compactness and the HeineBorel Theorem. 

15 
Chapter 18 (time permitting) 
Complete metric spaces. 
